water
Article Analytical Solutions for Unsteady Groundwater Flow in an Unconfined Aquifer under Complex Boundary Conditions
Yawen Xin , Zhifang Zhou *, Mingwei Li and Chao Zhuang
School of Earth Science and Engineering, Hohai University, Nanjing 210098, China; [email protected] (Y.X.); [email protected] (M.L.); [email protected] (C.Z.) * Correspondence: [email protected]
Received: 23 October 2019; Accepted: 21 December 2019; Published: 24 December 2019
Abstract: The response laws of groundwater dynamics on the riverbank to river level variations are highly dependent on the river level fluctuation process. Analytical solutions are widely used to infer the groundwater flow behavior. In analytical calculations, the river level variation is usually generalized as instantaneous uplift or stepped, and then the analytical solution of the unsteady groundwater flow in the aquifer is derived. However, the river level generally presents a complex, non-linear, continuous change, which is different from the commonly used assumptions in groundwater theoretical calculations. In this article, we propose a piecewise-linear approximation to describe the river level fluctuation. Based on the conceptual model of the riverbank aquifer system, an analytical solution of unsteady groundwater flow in an unconfined aquifer under complex boundary conditions is derived. Taking the Xiluodu Hydropower Station as an example, firstly, the monitoring data of the river level during the period of non-impoundment in the study area are used to predict the groundwater dynamics with piecewise-linear and piecewise-constant step approximations, respectively, and the long-term observation data are used to verify the calculation accuracy for the different mathematical models mentioned above. During the reservoir impoundment period, the piecewise-linear approximation is applied to represent the reservoir water level variation, and to predict the groundwater dynamics of the reservoir bank.
Keywords: complex boundary conditions; piecewise-linear approximation; piecewise-constant step approximation; analytical solutions; unsteady groundwater flow
1. Introduction The interaction between groundwater and surface water is a universal phenomenon in the natural world and an important part of the terrestrial hydrological cycle [1,2]. In the case of hydraulic connections between surface rivers and unconfined water, the river level variation is the key factor affecting the groundwater dynamics on both sides of the river [3–5]. When the river level rises, surface water infiltrates into the groundwater. This can lead to environmental and geological problems [2,4,6–8] such as solute transport, groundwater pollution, reservoir immersion, and land salinization. Therefore, understanding surface–subsurface water interactions and being able to compute the effect of these interactions are essential to address environmental and geological problems. Analytical methods and numerical simulations are often used for studying the unsteady groundwater flow caused by river level variations. In general, the numerical simulation method requires complicated algorithms to solve the problem of convergence and conservation of mass for specific models [9–11] and to obtain an approximate solution. The analytical solution needs to obey strict assumptions [12]. However, it has intuitive and concise expression, and it is relatively easy to
Water 2020, 12, 75; doi:10.3390/w12010075 www.mdpi.com/journal/water Water 2020, 12, 75 2 of 13 obtain unsteady groundwater flow laws and give an effective approach to evaluate the calculation accuracy. For simplified mathematical models, the commonly used analytical solutions are usually based on the condition that the river level remains constant after the instantaneous change has been completed. In addition, the J.G. Ferris model [13] is a classical method to solve such problems. However, the actual change of the river level is different from the assumptions in groundwater flow theoretical calculations, which are frequently affected by many factors, mainly depending on the topographic, geomorphological, hydrological, and meteorological conditions of the basin; the hydraulic characteristics of the basin itself; and the influence of artificial regulation. Meanwhile, the river level variation has its own characteristics: Firstly, the water level varies nonlinearly and continuously with time; secondly, the water level presents periodic variation; thirdly, the water level varies sharply in the wet season and gently in the dry season. For mathematical simplicity, the piecewise-constant step approximation is the most commonly used approach to represent the river level variation, and then such a problem is analyzed by the superposition principle combined with the solution of the classical model [14]. However, when the river level variation is more complex, the piecewise-constant step approach needs more segments to describe the variation process, and the calculation is more cumbersome. Moreover, the commonly used approach to represent the river level variation may not always be sufficient to capture important details in the observed hydraulic transient and continuous variation characteristics, which will also result in great errors to the unsteady groundwater flow calculation. For cases where the river level variability exhibits increasing or decreasing linear trends, a step-change approximation is generally not suitable unless a large number of closely spaced step changes are introduced. Therefore, scientific and reasonable approximations to boundary conditions will directly affect the solution accuracy. For that, we can be aided by the use of special functions that are not commonly used, or we can construct special functions to represent boundary conditions [12,14–17]. The piecewise-linear approximation approach is widely used in solving nonlinear problems [18–22], and it is an ideal approach for representing time-varying functions. Songtag [23] considers that piecewise-linear approximation can supervise nonlinear problems; Zhuang [21] derives an analytical solution to estimate the hydraulic parameters in an aquitard by a piecewise-linear approximation of long-term observed water level data in adjacent aquifers and analyzes the aquitard hysteresis characteristics. Piecewise-linear approximation of the pumping rate variable has been illustrated in the petroleum engineering literature (Stewart [20]). Phoolendra [19] applies piecewise-linear approximation to represent the sinusoidally varying pumping test, and the close correspondence between analytically computed drawdowns with and without piecewise-linear approximations validates the proposed methodology. In this article, the piecewise-linear approximation is extended and applied to the generalization of the river level variation. Based on the condition that the river boundary is on one side of the model and the impervious boundary is on the other side and the flow is considered to be in a stable state at the initial time, the analytical solution of unsteady flow in the unconfined aquifer under complex boundary conditions is derived. It is noteworthy that the analytical solution proposed in this study takes into account the thickness variation in the unconfined aquifer. At the same time, the river level is generalized by the commonly used method of piecewise-constant step approximation, and the corresponding analytical solution is also derived. Taking the Xiluodu Hydropower Station as an example, two methodologies (piecewise-linear and piecewise-constant step approximations) are used to calculate water level variations in an unconfined aquifer caused by the river level variation. The results are compared with those for the long-term observed data to illustrate the calculation accuracy of the methodologies, which demonstrated that the piecewise-linear approximation has less relative error. After validating our methodology, we analyze the reservoir impoundment process by the piecewise-linear approximation and predict the unsteady groundwater flow in an unconfined aquifer caused by reservoir impoundment. Water 2020, 12, x FOR PEER REVIEW 3 of 14 error. After validating our methodology, we analyze the reservoir impoundment process by the piecewise-linear approximation and predict the unsteady groundwater flow in an unconfined aquifer caused by reservoir impoundment.
2. Theory and Methodology Water 2020, 12, 75 3 of 13
2.1. Linearization and Solution of the Groundwater Flow Model near the Riverbank 2. TheoryDupuit and’s assumption Methodology and Boussinesq’s equation play important roles in solving groundwater flow in unconfined aquifers and are still widely adopted in hydrogeological calculations. In order to 2.1. Linearization and Solution of the Groundwater Flow Model near the Riverbank study the unsteady groundwater flow under river boundary conditions, a conceptual model of groundDupuit’swater flow assumption near the and river Boussinesq’s bank is equationestablished play, as important shown in roles Figure in solving 1, with groundwater the following flow assumptions:in unconfined (a) aquifers the unconfined and are still aquifer widely is homogeneous adopted in hydrogeological and isotropic, calculations.where the impermeable In order to studybase isthe horizontal unsteady and groundwater the upper flowinfiltration under rivercan be boundary neglected conditions,; (b) the groundwater a conceptual flow model diverted of groundwater by rivers canflow be near regarded the river as one bank-dimensional is established,, and as the shown flow in in Figure the unconfined1, with the aquifer following follows assumptions: Darcy’s law (a); the(c) tunconfinedhe initial hydraulic aquifer ishead homogeneous in the unconfined and isotropic, aquifer where is uniform the impermeable and is consistent base iswith horizontal the river and level the; (d)upper the infiltration boundary conditions can be neglected; of the (b)established the groundwater model follow flow divertedthat the byleft rivers side is can the be river regarded’s level as boundary,one-dimensional, and the and right the side flow is in the the impervious unconfined boundary aquifer follows, considering Darcy’s the law; study (c) the area initial as hydraulica certain distancehead in the from unconfined the valley aquifer rather is uniform than on and the is whole consistent stratigraphic with the river scale. level; The (d) corresponding the boundary mathematicalconditions of themodel established can be expressed model follow as: that the left side is the river’s level boundary, and the right side is the impervious boundary, considering the study area as a certain distance from the valley rather than on the whole stratigraphic scale. The corresponding mathematical model can be expressed as: K h h (H ) (0 x l,t 0) K ∂x ∂hx ∂ht H = (0(0 x xl) l, t 0) hµ(x∂,xt) ∂x h(x,∂0t) t 0 ≤ ≤ ≥ (1) h(x, t) t=0 = h(x, 0) (t (00) < x < l) h(x,t) h(x,0) g(t) (1) x 0 ( ) h(x, t) x=0 = h(x, 0) + g(t) (t 0) t > 0 h (x,t) 0 hxx(x, t)x=l = 0 (t > 0) |x l wherewhere KK isis the the hydraulic hydraulic conductivity, conductivity, µµ isis the the specific specific yield, yield, hh isis the the water water level, level, HH isis the the aquifer aquifer thickness,thickness, HH = h, x isis thethe horizontal horizontal distance, distance,t ist theis the time, time, and andg(t )g is(t the) is functionthe function of the of boundary the boundary water waterlevel variationlevel variation with time.with time.
FigureFigure 1. 1. ConceptualConceptual model model of of groundwater groundwater flow flow near near the the riverbank. riverbank.
WhenWhen hh(x(x,t,)t )− hh(x(x,0),0) ≤ 0.10.1 hmh m(dividing(dividing the the whole whole calculation calculation time time t intot into several several time time units, units, hm hism − ≤ theis the aquifer aquifer average average thickness thickness in the in thetime time unit, unit, which which can can be satisfied be satisfied in many in many practical practical problems problems in anin anunconfined unconfined aquifer aquifer),), we we can can apply apply the the first first linear linear method method of of the the Boussinesq’ Boussinesq’ss equation equation to to the the mathematicalmathematical model, model, where where uu isis the the water water level level variation, variation, uu((xx,,tt)) == hh((xx,t,t)) − hh(x(x,0,0),), aa isis the the hydraulic hydraulic − diffusivitydiffusivity in in unconfined unconfined aquifer, aquifer, a = KKhhmm/µ/µ, ,and and E Equationquation (1) (1) can can be be transformed transformed into: into:
∂2u ∂u a 2 = (0 x l, t 0) ∂x ∂t ≤ ≤ ≥ u(x, t) = 0 (0 < x < l) t=0 (2) | u(x, t) x=0 = g(t) (t > 0) ux(x, t) = = 0 (t > 0) |x l Water 2020, 12, 75 4 of 13 the general solution of the mathematical model is given as:
( ) = ( ) + P∞ ( ) 2n 1 + P∞ ( ) 2n 1 u x, t g t Gn t sin 2−l πx Hn t sin 2−l πx n=1 n=1 4 R t ∂g 2n 1 2 Gn(t) = exp a − π (t τ) dτ (3) − (2n 1)π 0 ∂τ · − 2l − − 2 ( ) = 4 ( ) 2n 1 Hn t (2n 1)π g 0 exp a 2−l π t − − · − 2.2. Simple Representation of the Piecewise-Linear Approximation for River Level Boundary The river level variation history is approximated as linear segments composed of n stages, considering that the river level history is recorded as g0, g1, g2, ... , gn at discrete time intervals t0, t1, t2, ... , tn, and t0 = 0. Expressing the river level variation as a piecewise-linear function allows writing g(t) as: i 1 X− ( ) = ( ) + ( )( ) g t βi t ti 1 βj tj tj 1 Hti 1 Hti (4) − − − − − − j=1 where ti is the time turning point at the junction between the ith and the (i + 1)th linear stage of the river level variation history, and βi = (gi gi 1)/(ti ti 1) is the slope of ith linear river level variation − − − − element. If βi = 0, the ith stage of the river level variation is zero, and if βi = βi+1 for all i values, the n stages of river level variation reduce to one stage under a constant variation slope. Hti is a unit step function, which equals one when t t and remains zero elsewhere. ≥ i The piecewise-linear approximation can not only express the general functions, which change regularly with time, such as sinusoidal function, exponential function, and so on, but they can also approximate the random variation of the river level with time, which is difficult to express in the general function form, and it can be generalized to different degrees according to the required accuracy. Shown as in Figure2, the boundary river level varies with time and presents di fferent sinusoidal, g1(t) = 2.0 + sin(30t/π), exponential, g2(t) = 1.2exp(0.2 + 0.08t), and random, g3(t), characteristics, approximated by piecewise-linear approximation to 19, 4, and 20 segments, respectively, which reflect the variation characteristics of the original data and the key points. The number of linear segments is automatically divided by the program according to the input accuracy requirements, not manually partitioned. The higher the accuracy requirement, the more linear segments are divided. The piecewise-linear approximation can also be applied to the tidal variation and variable-rate pumping tests. Water 2020, 12, x FOR PEER REVIEW 5 of 14
Figure 2. River level variations for sinusoidal (g1), exponential (g2), and random (g3) characteristics Figure 2. River level variations for sinusoidal (g1), exponential (g2), and random (g3) characteristics with piecewise-linearpiecewise-linear approximation.
g(t) is a continuous continuous function with respect to t, but its its derivative derivative with with respect respect to t isis discontinuous discontinuous at t thehe turning point ti.. In In order order to to avoid avoid the the phenomena phenomena of of sharp sharp breakpoints breakpoints in in the the definition definition domain domain caused by this problem, we have to derive the water level variation distribution in the unconfined aquifer sequentially, namely making the final water level variation distribution in the (i − 1)th stage as the initial condition for the ith stage. Using the variable separation approach the analytical solution for the initial-boundary value problem of Equations (1) to (4) is derived as:
i1 u(x,t) (t t ) (t t ) i i1 j j j1 j1
2n 1 2 i exp[a( ) (t t )(i j)] 16l 2 1 2l j 2n 1 j sin x (5) a 3 (2n 1)3 2n 1 2l j1 n1 exp[a( )2 (t t )] j1 2l
0 ( j i) (i j) 1 ( j i) (6) When the river level changes, the infiltration of river water into the bank slope is regarded as a process of saturated groundwater flow, and the unconfined aquifer thickness changes at any time. In order to give consideration to the calculation simplicity and the variation of actual unconfined aquifer thickness, the unconfined aquifer thickness is regarded as a constant value in the calculated time unit, and the initial unconfined aquifer thickness is taken as the difference between the river level and the impermeable base before the calculated time point. Following, the unconfined aquifer thickness in the next time unit is the sum of the initial thickness and the average water level rise in the previous time unit based on the iterative methods. Therefore, the mathematical model proposed in this paper takes into account the unconfined aquifer thickness variation for discrete time steps when calculating the unsteady groundwater flow. The average water level rise value is the average hydraulic response of the unconfined aquifer in the horizontal direction, which is achieved by integrating u(x,t) in Equation (5) in the horizontal direction and dividing it by the horizontal length L. The analytical solution of the average hydraulic response in the piecewise-linear approximation is as follows:
Water 2020, 12, 75 5 of 13 caused by this problem, we have to derive the water level variation distribution in the unconfined aquifer sequentially, namely making the final water level variation distribution in the (i 1)th stage as − the initial condition for the ith stage. Using the variable separation approach the analytical solution for the initial-boundary value problem of Equations (1) to (4) is derived as:
i 1 P− u(x, t) = βi(t ti 1) + βj(tj tj 1) − − j=1 − − − 2 (5) i exp[ a 2n 1 π (t t ) φ(i j)] 16l2 P P∞ 1 2−l j 2n 1 βj − 2 − · − − sin − πx aπ3 (2n 1)3 2n 1 2l j=1 n=1 − exp a − π (t tj 1) − 2l − − 0 (j = i) φ(i j) = (6) − 1 (j , i) When the river level changes, the infiltration of river water into the bank slope is regarded as a process of saturated groundwater flow, and the unconfined aquifer thickness changes at any time. In order to give consideration to the calculation simplicity and the variation of actual unconfined aquifer thickness, the unconfined aquifer thickness is regarded as a constant value in the calculated time unit, and the initial unconfined aquifer thickness is taken as the difference between the river level and the impermeable base before the calculated time point. Following, the unconfined aquifer thickness in the next time unit is the sum of the initial thickness and the average water level rise in the previous time unit based on the iterative methods. Therefore, the mathematical model proposed in this paper takes into account the unconfined aquifer thickness variation for discrete time steps when calculating the unsteady groundwater flow. The average water level rise value is the average hydraulic response of the unconfined aquifer in the horizontal direction, which is achieved by integrating u(x,t) in Equation (5) in the horizontal direction and dividing it by the horizontal length L. The analytical solution of the average hydraulic response in the piecewise-linear approximation is as follows:
R l u(x,t)dx u(t) = 0 = l 2n 1 2 i 1 i exp a − π (t tj) φ(i j) (7) P− 32l2 P P∞ 1 − 2l − · − − βi(t ti 1) + βj(tj tj 1) 4 βj 4 2 − − − − − aπ (2n 1) 2n 1 j=1 j=1 n=1 − exp a − π (t tj 1) − 2l − −
2.3. Simple Representation of the Piecewise-Constant Step Approximation for River Level Boundary In previous studies, the continuous variation of the river level is usually approximated to be a piecewise-constant step variation, that is, the river level in each period is regarded as a fixed value, and the river level variation between adjacent periods is considered to be instantaneous return water. The variable value of the river level is defined as H0, H1, H2, ... , Hn at time turning point t0, t1, t2, ... , tn, and t0 = 0. Then, the piecewise-constant step variation of river level with time can be expressed as:
g(t) = Hi ti 1 < t < ti (8) − the corresponding analytical solution of the piecewise-constant step approximation is given as:
i X∞ n X 2n 1 2 4 2 1 a( − π) (t tj 1) u(x, t) = Hi sin − πx (Hj Hj 1)e− 2l − − (9) − (2n 1)π 2l · − − n=1 − j=1 Water 2020, 12, 75 6 of 13 and the analytical solution of the average hydraulic response in the piecewise-constant step approximation is as follows:
R l i u(x, t)dx X∞ X 2n 1 2 0 8 a( − π) (t tj 1) u(t) = = Hi (Hj Hj 1)e− 2l − − (10) l − (2n 1)2π2 − − n=1 − j=1 In this case, the hydraulic diffusivity a in the u(x,t) is variable, that is, the unconfined aquifer thickness is a variable, which is determined by the initial thickness and the average hydraulic response value. As shown from the u(x,t) expressions, the series term exists both in piecewise-linear or piecewise-constant step approximations for river level variation. The series term indicates that when the boundary water level changes, the variation range of the water level at each point in an unconfined aquifer is not equal to the boundary water level variation, namely damping. Theoretically, the specific yield value of unconfined aquifer is usually 3–4 orders of magnitude larger than the specific storage value of the confined aquifer, so the groundwater flow in the unconfined aquifer is dominated by gravity conduction. Differing from the pressure conduction in confined aquifers, the water release caused by the river level rise in an unconfined aquifer is a process that is not instantaneous, which, depending on the progression of the saturated zone and leading to the hydraulic response, is relatively slow. When the boundary water level rises, the water level at each point in the unconfined aquifer may be lower than that in a period of time. The groundwater level reacts with delay towards the river level variation, that is to say, there is a time lag. In addition, the phenomenon is more obvious, especially when the boundary water level changes sharply. Considering that the impermeable boundary is far from the riverbank, the limit value of the rise in groundwater level in the unconfined aquifer will not exceed the maximum variation of the river level, which is completely consistent with the results of the analytical solution.
3. Field Application
3.1. Background Take the Xiluodu Hydropower Station as an example. The Xiluodu Hydropower Station is located on the main stream of Jinsha River in Leibo County of Sichuan Province and Yongshan County of Yunnan Province. The location of the study area is shown in Figure3. The reservoir has a normal water level of 600 m. The Xiluodu Hydropower Station cooperates with the Three Gorges Project to improve the flood control capacity of the middle and lower reaches of the Yangtze River, which give full play to the comprehensive benefits of the Three Gorges Project. Besides, it promotes the development of the western region and realizes the sustainable development of the national economy. The river bedrock and the valley slopes on both sides of the dam site are composed of Emeishan basalt (P2β) of the Upper Permian, and the Maokou limestone (P1m) of the Lower Permian is buried 70 m below the dam foundation. The research suggests that the patterns of valley and foundation deformation caused by reservoir impoundment processes are closely related to the groundwater flow in the aquifers. Therefore, it is of great significance to calculate the influence of reservoir impoundment on groundwater level [24]. Water 2020, 12, x FOR PEER REVIEW 7 of 14
groundwater level in the unconfined aquifer will not exceed the maximum variation of the river level, which is completely consistent with the results of the analytical solution.
3. Field Application
3.1. Background Take the Xiluodu Hydropower Station as an example. The Xiluodu Hydropower Station is located on the main stream of Jinsha River in Leibo County of Sichuan Province and Yongshan County of Yunnan Province. The location of the study area is shown in Figure 3. The reservoir has a normal water level of 600 m. The Xiluodu Hydropower Station cooperates with the Three Gorges Project to improve the flood control capacity of the middle and lower reaches of the Yangtze River, which give full play to the comprehensive benefits of the Three Gorges Project. Besides, it promotes the development of the western region and realizes the sustainable development of the national economy. The river bedrock and the valley slopes on both sides of the dam site are composed of Emeishan basalt (P2β) of the Upper Permian, and the Maokou limestone (P1m) of the Lower Permian is buried 70 m below the dam foundation. The research suggests that the patterns of valley and foundation deformation caused by reservoir impoundment processes are closely related to the Watergroundwater2020, 12, 75 flow in the aquifers. Therefore, it is of great significance to calculate the influence7 of of 13 reservoir impoundment on groundwater level [24].
Figure 3. Map showing the geographic location of the study area.
3.2. Representation of River Level In thethe analyticalanalytical solution,solution, itit is is necessary necessary to to generalize generalize the the river river level level in in di differentfferent forms forms in in order order to calculateto calculate the theunsteady unsteady flow flow in the in unconfined the unconfined aquifer aquifer caused caused by the river by the level river variation level variation and compare and thecompare piecewise-linear the piecewise model-linear with model the traditional with the traditional piecewise-constant piecewise step-constant model step proposed model in proposed this article. in Considerthis article the. Consider water level the water monitoring level monitoring data of Jinsha data Riverof Jinsha in1995 River as in original 1995 as dataoriginal tofit. data As to shown fit. As inshown Figure in4 Figure, the river 4, the level river variation level variation in a hydrological in a hydrological year presents year present seasonals seasonal variation. variation. The rise The of riverrise of level river is level concentrated is concentrate in June–September,d in June–September, during during which which there isthere obvious is obvious fluctuations fluctuation of waters of levelwater likely level relatedlikely related to seasonal to seasonal rainfall. rainfall On the. On whole, the whole, the water the water level level variation variation is complex, is complex, and and it is diit ffiis cultdifficult to express to express its change its change with simple with simple functions. function The rivers. The level river is dividedlevel is intodivided 7 and into 17 segments7 and 17 bysegments piecewise-linear by piecewise and-linear piecewise-constant and piecewise step-constant approximations. step approximations. The more segments The more are segments divided, theare closerdivided, the theapproximate closer the values approximate are to the original values data.are to The the time original segments data. are The automatically time segments divided are by theautomatically written program divided according by the written to the input program accuracy according requirements, to the input not accuracy manually requirements, partitioned. The not piecewise-linearmanually partitioned. approximation The piecewise proposed-linear in this approximation paper is used proposed to generalize in this the paperriver level, is used which to cangeneralize not only the reflect river level, the water which level can changenot only trends reflec butt the also water show level the change characteristic trends but water also levelshow and the correspondingcharacteristic water time points. level and However, corresponding the traditional time points. piecewise-constant However, the step traditional approximation piecewise can- only represent the overall change characteristics and ignores the stage change characteristics. This demonstrates that, for the cases where river level variations are better represented by piecewise-linear approximation, as river level variations may not be adequately expressed by a relatively small number of step segments, the piecewise-constant step approximation for representing river level transients is not always sufficient to accurately represent the observed data. Water 2020, 12, x FOR PEER REVIEW 8 of 14
constant step approximation can only represent the overall change characteristics and ignores the stage change characteristics. This demonstrates that, for the cases where river level variations are better represented by piecewise-linear approximation, as river level variations may not be adequately expressed by a relatively small number of step segments, the piecewise-constant step approximation Waterfor representing2020, 12, 75 river level transients is not always sufficient to accurately represent the observed8 of 13 data.
FigureFigure 4.4. RiverheadRiverhead variationvariation inin 19951995 isis divideddivided intointo 77 segmentssegments ((aa)) andand 1717 segmentssegments ((dd),), respectivelyrespectively withwith piecewise-linearpiecewise-linear andand piecewise-constantpiecewise-constant stepstep approximations;approximations; ii isis thethe segments;segments; andand calculatedcalculated resultsresults ofof groundwater-levelgroundwater-level variationvariation forfor long-termlong-term observationobservation wellswells X35X35 ((bb,,ee)) andand X50X50 ((cc,,ff)) withwith piecewise-linearpiecewise-linear andand piecewise-constantpiecewise-constant stepstep approximationsapproximations for for di differentfferent segments. segments.
3.3.3.3. ModelModel CalculationCalculation InIn orderorder toto studystudy thethe unsteadyunsteady flowflow ofof thethe unconfinedunconfined aquiferaquifer causedcaused byby the river level variation, itit isis necessary necessary to to analyze analyze and and determine determine the the hydrogeological hydrogeological parameters parameters of the of basalt the basalt in the in bank the slope.bank Reasonableslope. Reasonable values of values the hydrogeological of the hydrogeological parameters parameters are very important are very for important the reliable for analysis the reliab of thele calculationanalysis of results.the calculation The hydraulic results. conductivity The hydraulic of the conductivity basalt area measuredof the basalt by fieldarea tests measured of the Xiluoduby field Hydropowertests of the Xiluodu Station Hydropower ranged from 0.06Station m/ drange to 2.70d from m/d, 0.06 in which m/d to the 2.70 hydraulic m/d, in conductivity which the hydraulic and the porosityconductivity of rock and mass the were porosity small. of rock The average mass were hydraulic small. conductivity The average and hydraulic porosity conductivity (K = 0.90 m and/d, nporosity= 0.03) after(K = excavation0.90 m/d, n were = 0.03) selected after excavation to calculate were and specificselected yield to cal wasculate approximated and specific as yield porosity. was Theapproximated initial thickness as porosity. of the unconfined The initial aquifer thickness was ofH the= 153 unconfined m, and the aquifer horizontal was extension H = 153 m, distance and the of thehorizontal study area extension was 7.9 distance km. of the study area was 7.9 km. At 85.7 m and 371.4 m away from the river valley, the groundwater level at the selected location was calculatedAt 85.7 m and by piecewise-linear 371.4 m away from and the piecewise-constant river valley, the groundwater step models, level respectively. at the selected At the location same time,was calculated the dynamic by observedpiecewise data-linear of theand water piecewise level at-constant the corresponding step model long-terms, respectively. observation At the wellssame X35time, and the X50 dynamic were usedobserved to verify data the of the accuracy water oflevel the at two the models. corresponding The calculated long-term results observation are shown wells in FigureX35 and4. ItX50 is obviouswere used from to verify the fitting the accuracy degree of of the the calculated two models. results The andcalculated the long-term results are observation shown in dataFigure that 4. theIt is groundwater obvious from level the fitting calculated degree by of the the piecewise-linear calculated results model and wasthe long closer-term to the observation dynamic observeddata that datathe groundwater of the long-term level observation calculated by well. the In piecewise order to- quantifylinear model the calculation was closer accuracyto the dynamic of the twoobserved models, data the of RMSEthe long (root-term mean observation square error) well. andIn order RE (relative to quantify error) the were calculation used to accuracy represent of thethe deviationtwo models, between the RMSE the calculated (root mean and square the observed error) and values, RE (relative which could error) be were expressed used to with represent Equations the (11)deviation and (12) between and the the calculated calculated results and the as shownobserved in Tablevalues1., which The error could analysis be expressed showed with that Equation the mores segments(11) and (12 were) and divided, the calculated the higher results the calculationas shown in accuracy Table 1. was,The error and theanalysis calculation showed accuracy that the of more the piecewise-linear model was higher than that of the piecewise-constant step model. v t n 1 X RMSE = (H H )2 (11) n c,i − m,i i=1 Water 2020, 12, 75 9 of 13
n 1 X Hc,i Hm,i RE = − 100% (12) n Hm i × i=1 , where Hc,i denotes the calculation value of groundwater level at the ith time point, Hm,i denotes the corresponding monitor data, and n is the total number monitored.
Table 1. Root-Mean-Square Error (RMSE) and Relative-Error (RE) between observed and calculated data with Piecewise-Linear (Columns 3 and 4) and Piecewise-Constant Step (Columns 5 and 6).
Linear Approximation Step Approximation RMSE RE RMSE RE X35 1.74 0.0033 2.33 0.0046 i = 7 X50 2.35 0.0047 2.63 0.0053 X35 1.50 0.0028 1.81 0.0036 i = 17 X50 1.82 0.0036 2.52 0.0049
In general, the groundwater dynamic in the long-term observation well was basically consistent with the river level dynamic, which represents good synchronization. The main reason is that the distance between the selected long-term observation well and the river valley was relatively short compared with the whole study area, and the signal damping phenomenon was not obvious. However, the long-term observation wells X35 and X50, which are located close to and far from the river valley, respectively, performed different response degrees of groundwater level to river level variation. Compared with the river level values and the groundwater level observed data, with the distance increasing, the amplitude of groundwater level variation decreased, and the groundwater level variation delay phenomenon in the X50 was more notable than that in the X35, which is related to the flow resistance of the rock. From the calculation results, the errors of X50 were greater than those of X35, which may be due to the selection of hydraulic parameters. For the convenience of model calculation, we chose the same hydraulic diffusivity in an unconfined aquifer to calculate the groundwater dynamic at different locations from the valley. However, the hydraulic diffusivity in an unconfined aquifer near a river valley is generally larger than that far from valleys, which is related to the good permeability of rock masses and the fast discharge of groundwater near the river valley. Therefore, the same hydraulic diffusivity in the unconfined aquifer selected may affect the accuracy of groundwater level calculations at different locations.
3.4. Prediction of Groundwater Level Fluctuation Caused by Reservoir Water Level Variation From the model calculation results and groundwater dynamic data in the long-term observation well, the piecewise-linear model has a high reliability in calculating groundwater flow caused by the boundary water level variation. This example has shown that the analytical model can be used to represent the flow with sufficient accuracy. The Xiluodu Hydropower Station has been impounded since 30 October 2012. The initial river level was 383.45 m. In November 2014, the impoundment had reached the normal water level of 600 m, and in the observed data of the reservoir water level until 26 February 2018, the water level fluctuated periodically. Taking the initial time of 30 October 2012 and the end time of 26 February 2018 for 1943 days, the reservoir water level variation was generalized by a piecewise-linear approximation, which was divided into 43 continuous linear segments, as shown in Figure5. Water 2020, 12, x FOR PEER REVIEW 10 of 14
and the end time of 26 February 2018 for 1943 days, the reservoir water level variation was Watergeneralized2020, 12, 75 by a piecewise-linear approximation, which was divided into 43 continuous 10 linear of 13 segments, as shown in Figure 5.
Figure 5. Reservoir water level duration curve represented withwith piecewise-linearpiecewise-linear approximation.approximation.
We analyzed analyzed the the unsteady unsteady flow flow of unconfined of unconfined aquifer aquifer caused caused by the reservoirby the reservoir water level water variation level accordingvariation according to the piecewise-linear to the piecewise approximation-linear approximation proposed in proposed this paper. in Fourthis paper. selected Four time selected nodes weretime 9nodes February were 2013,9 February 16 March 2013, 2014, 16 Ma 29rch July 2014, 2015, 29 and July 14 2015, February and 14 2018 February corresponding 2018 corresponding to 100, 500, to 1000, 100, and500, 19001000 d, and (based 1900 on d the (based initial on time the ofinitial impoundment) time of impoundment) to analyze the to distribution analyze the of distribution the water level of the at eachwater point level in at the each study point area. in the At thestudy same area. time, At groundwaterthe same time, level groundwater variations withlevel timevariatio werens considered with time atwere 0.2, considered 0.5, 1.0, 2.0, at 3.0, 0.2, 5.0, 0.5, and 1.0, 7.9 2.0, km 3.0, (the 5.0 water, and 7.9 barrier km (the boundary) water barrier away from boundary) the reservoir away from bank, the as shownreservoir in bank Figure, as6. shown A rising in reservoir Figure 6. water A rising level reservoir will recharge water the level unconfined will recharge aquifer the and unconfined cause a long-termaquifer and and cause wide-ranging a long-term impact. and The wide explanation-ranging of impact. the long-term The explanation characteristic of is the that long basalt-term has goodcharacteristic integrity, ais large that thickness, basalt has a compact good integrity, structure, lowa large permeability, thickness, and a most compact are dry structure under natural, low conditions.permeability, In and addition, most are the dryhydraulic under responsenatural conditions. of groundwater In addition, in the unconfined the hydraulic aquifer response mainly of dependsgroundwater on gravity, in the supplemented unconfined aquifer by pressure mainly transfer. depend Whens on river gravity, water supplemented recharges unconfined by pressure water intransfer. basalt strata,When the river developed water recharges joints, interlayer, unconfined and water internal in shear basalt zones strata, are the saturated developed firstly. joints The, processinterlayer, ofmountain and inter saturationnal shear is zones extremely are saturated slow, and firstly the corresponding. The process water of mountain level rise is saturat slow, soion the is riverextremely water slow, will recharge and the thecorresponding unconfined aquiferwater level for a rise long is time. slow, The so the wide river range water was will due recharge to the broad the horizontalunconfined extension aquifer for of basalta long stratumtime. The and wide the low,range gentle, was due natural to the groundwater broad horizontal level of extension unconfined of water.basalt stratum What is and more, the after low, reservoir gentle, natural impoundment, groundwa thereter level was of a largeunconfined distribution water. area, What where is more, the unconfinedafter reservoir surface impoundment, elevation of thethere mountain was a bodylarge was distribution smaller than area the, where reservoir’s the unconfined normal water surface level. elevationAt the of same the mountain time, the body water was level smaller decreased than graduallythe reservoir’ as its extendednormal water from leve thel reservoir. bank to the side away from that, showing obvious damping of gravity water release. The unconfined water At the same time, the water level decreased gradually as it extended from the reservoir bank to surface advanced continuously as time went on. When the time was 100 d, the transfer distance of the side away from that, showing obvious damping of gravity water release. The unconfined water groundwater level fluctuation caused by the reservoir water level variation was 4200 m, which is surface advanced continuously as time went on. When the time was 100 d, the transfer distance of shown in the A point coordinates (4200, 383.45); when the time was 500 d, it had been transferred to groundwater level fluctuation caused by the reservoir water level variation was 4200 m, which is the impervious boundary, where the water level was 383.56 m; when the time was 1000 d, the water shown in the A point coordinates (4200, 383.45); when the time was 500 d, it had been transferred to level at the impervious boundary was 390.33 m; when the time was 1900 d, the water level at the the impervious boundary, where the water level was 383.56 m; when the time was 1000 d, the water impervious boundary is 434.25 m. The analyzed data show that the water level rises slowly and never level at the impervious boundary was 390.33 m; when the time was 1900 d, the water level at the equals the reservoir water level, which indicates that the unconfined aquifer response to the reservoir impervious boundary is 434.25 m. The analyzed data show that the water level rises slowly and never water level variation has obvious damping characteristics. At time 0 d, the groundwater level is at equals the reservoir water level, which indicates that the unconfined aquifer response to the reservoir the initial time. According to the existing survey data, at the initial time, the fluctuation range of the water level variation has obvious damping characteristics. At time 0 d, the groundwater level is at groundwater level in the whole study area was very small, only 0.5 m, so the groundwater level at this the initial time. According to the existing survey data, at the initial time, the fluctuation range of the time was approximately a plane, with a value of 383.45 m. groundwater level in the whole study area was very small, only 0.5 m, so the groundwater level at Comparing the time-varying curves of groundwater levels at 0.2, 0.5, 1.0, 2.0, 3.0, 5.0, and 7.9 km this time was approximately a plane, with a value of 383.45 m. away from the reservoir bank, the groundwater level at the same location showed an overall upward trendComparing with time. the Zero time kilometer-varying indicates curves of the groundwater reservoir water level levels at 0.2, variation. 0.5, 1.0, At2.0, 0.2, 3.0, 0.5, 5.0, and and 1.07.9 km,km theaway groundwater from the reservoir level fluctuated bank, the groundwater obviously with level time, at the showing same location a good show synchronizationed an overall with upward the reservoir water level variations, although the increase was less than that. At 2.0 km, there was slight groundwater level fluctuation. With the distance increasing, the fluctuation of the groundwater level Water 2020, 12, x FOR PEER REVIEW 11 of 14 trend with time. Zero kilometer indicates the reservoir water level variation. At 0.2, 0.5, and 1.0 km, Waterthe gro2020undwater, 12, 75 level fluctuated obviously with time, showing a good synchronization with11 of the 13 reservoir water level variations, although the increase was less than that. At 2.0 km, there was slight groundwater level fluctuation. With the distance increasing, the fluctuation of the groundwater level nearly disappears,disappears, reducedreducedgradually gradually at at 3, 3, 5 5 and and 7.9 7.9 km, km, showing showing only only an an increase increase of of the the water water level. level. In Inaddition, addition, when when the reservoir the reservoir water water level changed level change periodically,d periodically, the damping the damping of the hydraulic of the hydraulic response responsein the unconfined in the unconf aquiferined was alsoaquifer reflected was inalso the reflected phase di ff inerence the phasebetween difference the groundwater between level the groundwaterand the reservoir level water and level. the reservoir When the water reservoir level. water When level the changed reservoir from water ascending level change to descendingd from (descendingascending to todescending ascending), (descending the groundwater to ascending), level at athe certain groundwater position level away at from a certain the bank position slope away kept fromascending the bank (descending) slope kept for ascending a period (descending) of time, and then,for a itperiod began of to time, descend and (ascend),then, it began which to indicatesdescend (ascend),the existence which of aindicate certains phase the existence difference. of a The certain farther phase away difference. from the The bank farther slope, away the bigger from thethe phasebank slope,difference the is,bigger and itthe cannot phase even difference reflect theis, and fluctuations it cannot of even the reservoirreflect the water fluctuations level. It canof the be reservoir seen that waterthe groundwater level. It can levelbe seen changed that the greatly groundwater near the level bank change slope, andd greatly the hydraulic near the dampingbank slope, eff andect was the hydraulicsmaller, which damping can reflect effect the was change smaller characteristics, which can ofreflect the reservoir the change water characteristics level. However, of the the reservoir variation waterrange level. of groundwater However, the level variation was smaller range far of fromgroundwater the bank level slope, was the smaller hydraulic far from damping the bank effect slope, had thegreater hydraulic influence, damping and the effect change had characteristics greater influen ofce, the and reservoir the change water characteristics level attenuated of accordingly. the reservoir It wateris worth level to pointattenuate outd that accordingly. this system It behavioris worth to (phase point di outfference, that this damped system fluctuation) behavior (phase was in difference, line with dampedother studies fluctuation) [25]. Since wa thes in model line with is assumed other studies to be an [25] impervious. Since the boundary, model is the assumed limit value to beof the an impergroundwatervious boundary, level variation the atlimit each value point of in the unconfinedgroundwater aquifer level will variation not exceed at each the maximum point in rise the valueunconfined of the aquifer reservoir will water not level.exceed The the steady maximum state rise flow value of groundwater of the reservoir was notwater being level. achieved The steady after state2000 days.flow of groundwater was not being achieved after 2000 days.
Figure 6. 6. GroundwGroundwater-levelater-level fluctuation fluctuation curves curves at different at di pointsfferent in study points area in ( studya) and groundwater area (a) and- levelgroundwater-level duration curves duration (b) in the curves unconfined (b) in the aquifer. unconfined aquifer.
4. Results Results and and Discussions Discussions The analytical analytical solutions solutions of groundwater of groundwater level level versus versus time and time distance and distance are developed are developed to investigate to investigatethe impacts the of theimpacts boundary of the water boundary level approximation,water level approximation models accuracy,, models and accuracy, damping and eff ectsdamping in an effectsunconfined in an aquifer. unconfined The mathematicalaquifer. The modelmathematical assumes model that rainfall, assumes aquifer that rainfall, recharge, aquifer extractions, recharge, and extractions,base flow discharge and base areflow zero, discharge which are may zero, be diwhichfferent may from be thedifferent actual from situation. the actual This situation. paper mainly This paperdiscusses mainly the influence discusses of theboundary influence water of level boundary on groundwater water level flow. on In groundwater order to simplify flow. the In analytical order to simplifysolution, the other analytical recharge solution, and discharge other recharge values are and considered discharge as values zero. are considered as zero.
4.1. Effect Effect of Boundary Water Level Approximation Analytical solutions are derived to study the groundwater flowflow inin thisthis article.article. The analytical solutions, di dffifferenterent from from numerical numerical simulation, simulation, and the and boundary the boundary conditions conditions could not becould represented not be representedas discrete data as discrete points.However, data points. the However, original time the resolution original time of the resolution river level of versus the river time level series versus could timenot be series represented could not with be a represented normal function. with a Therefore, normal function. we applied Therefore, the piecewise-linear we applied approximationthe piecewise- linearto generalize approximation the river to level generalize variation, the so river it could level be variation shown with, so it an could expression. be shown Linear with segments an expression. (7 and 17) were divided in the river level, as shown in Figure4a,b, and the results indicated that the higher the time resolution, the smaller the deviation between the calculated values and the original data. Water 2020, 12, 75 12 of 13
4.2. Effect of Models Accuracy Figure4 displays the curves of groundwater level versus time for long-term observation wells X35 and X50 with piecewise-linear and piecewise-constant step approximations. Table1 shows the error analysis with different approximations. Both demonstrate that the accuracy of the piecewise-linear approximation was higher than that of the piecewise-constant step with the same time resolution.
4.3. Effect of Damping in the Unconfined Aquifer Figure6 presents the groundwater response lags behind the reservoir water level variation. The further away from the reservoir bank, the more obvious the time delay. It is obvious that the filling of the reservoir overrides this surface subsurface water interaction effect in the first 700 days. In addition, the groundwater level has not achieved a steady state, even after 2000 days, and the maximum water level at 7.9 km was far below the reservoir level in the simulation period. If the reservoir water level still changes periodically and the simulation time is extended, the water level at the impervious boundary could reach to the reservoir’s impoundment level at about 6000 days from the initial time, which is about the year 2030.
5. Conclusions The river level usually presents a non-linear continuous change over time, which is different from the instantaneous rise or piecewise-constant step variation of the water level. According to the river level variation characteristics, a piecewise-linear approximation is proposed to generalize the river level variation. Based on the actual situation, the other side of the study area is set as an impervious boundary, and the analytical solution of groundwater unsteady flow caused by the river level variation is derived. Results were compared with the traditional piecewise-constant step model to illustrate the calculation accuracy. Taking the Xiluodu Hydropower Station as an example to validate the method proposed in this paper, the piecewise-linear approximation values to the boundary water level are closer to the actual water level variation process, which can reflect the variation characteristics. The calculation accuracy of the piecewise-linear model is higher than that of the piecewise-constant step model through the verification of groundwater dynamic monitor data in the long-term observation well. The response of the groundwater level dynamic in an unconfined aquifer to the reservoir water level variation has obvious damping, which demonstrates that the farther away from the bank slope, the more obvious the time delay. The piecewise-linear approximation of the time-varying boundary water level can also be applied to predict the hydraulic response in the confined aquifers and aquitards. It should be made clear that this exercise is only relevant for the analytical solutions. Numerical models can only use the boundary condition time series on the resolution of the model. Based on the groundwater level variation obtained, we will estimate the surface-subsurface water exchange and study the influence of rainfall, aquifer recharge, extractions, and base flow discharge on the groundwater level and quantity in further work.
Author Contributions: Conceptualization, Y.X. and Z.Z.; methodology, Y.X.; software, Y.X., M.L. and C.Z.; validation, Z.Z., M.L. and C.Z.; formal analysis, Y.X.; investigation, Y.X.; resources, Z.Z.; data curation, Y.X.; writing—original draft preparation, Y.X.; writing—review and editing, Y.X. and Z.Z.; visualization, Y.X.; supervision, Z.Z. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China, grant number No. 41572209, and China Three Gorges Corporation, grant number No. 20188019316. Conflicts of Interest: The authors declare no conflict of interest.
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